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Page 1
Solutions of First-Order Differential Equations
Discipline Course – I
Semester - II
Paper: Differential Equations - I
Lesson: Solutions of First-Order Differential
Equations
Lesson Developer: Anu Sharma
College: Aditi Mahavidalaya, University of
Delhi
Institute of Lifelong Learning, University of Delhi pg. 1
Page 2
Solutions of First-Order Differential Equations
Discipline Course – I
Semester - II
Paper: Differential Equations - I
Lesson: Solutions of First-Order Differential
Equations
Lesson Developer: Anu Sharma
College: Aditi Mahavidalaya, University of
Delhi
Institute of Lifelong Learning, University of Delhi pg. 1
Solutions of First-Order Differential Equations
Table of Contents:
Chapter: Solutions of First-Order Differential Equations
• 1. Learning Outcomes
• 2. Introduction
• 3. Standard forms of First Order Differential Equations
o 3.1. Total Differential Function
o 3.2. Exact Differential Equation
• Exercise 1
• 4. Separable Equations
o 4.1. Solution of Separable Equations
• Exercises 2
• 5. Homogeneous Equations
o 5.1. Solution of Homogeneous Equations
• Exercises 3
• 6. Integrating Factors (I.F.)
• 7. Linear Equations
o 7.1. Solution of Linear Equation
o Exercise 4
o 7.2. Equations reducible to linear Form
o Exercises 5
• 8. Special Integrating Factors and Transformations
o 8.1. Rules for finding Integrating Factors
• Exercise 6
• 9. Special Transformations
o 9.1. Equations reducible to Homogeneous From
• Exercise 7
• Summary
• References
Institute of Lifelong Learning, University of Delhi pg. 2
Page 3
Solutions of First-Order Differential Equations
Discipline Course – I
Semester - II
Paper: Differential Equations - I
Lesson: Solutions of First-Order Differential
Equations
Lesson Developer: Anu Sharma
College: Aditi Mahavidalaya, University of
Delhi
Institute of Lifelong Learning, University of Delhi pg. 1
Solutions of First-Order Differential Equations
Table of Contents:
Chapter: Solutions of First-Order Differential Equations
• 1. Learning Outcomes
• 2. Introduction
• 3. Standard forms of First Order Differential Equations
o 3.1. Total Differential Function
o 3.2. Exact Differential Equation
• Exercise 1
• 4. Separable Equations
o 4.1. Solution of Separable Equations
• Exercises 2
• 5. Homogeneous Equations
o 5.1. Solution of Homogeneous Equations
• Exercises 3
• 6. Integrating Factors (I.F.)
• 7. Linear Equations
o 7.1. Solution of Linear Equation
o Exercise 4
o 7.2. Equations reducible to linear Form
o Exercises 5
• 8. Special Integrating Factors and Transformations
o 8.1. Rules for finding Integrating Factors
• Exercise 6
• 9. Special Transformations
o 9.1. Equations reducible to Homogeneous From
• Exercise 7
• Summary
• References
Institute of Lifelong Learning, University of Delhi pg. 2
Solutions of First-Order Differential Equations
1. Learning Outcome:
After reading this chapter, you should be able to :
(a) Recognize standard forms of First Order Differential Equations
(b)Exact Differential Equations and their solutions
(c)Separable Equations and equations reducible to this form and
their solutions
(d)Homogeneous Equations and equations reducible to this form
and their solutions
(e)Integrating Factors
(f)Linear Equations and Bernoulli’s Equations and their solutions
(g)Special Integrating Factors and Transformations
2. Introduction:
This chapter deals with first order differential equations for
which exact solutions can be obtained by definite methods. Here we
will discuss exact equations, separable equations, homogeneous
equations, linear equations and Bernoulli's equations. The aim of
this chapter is to recognize these various types of differential
equations and solve these by using various methods. Let us discuss
these equations one by one.
3. Standard forms of First Order Differential Equations
The differential equations of first order and first degree are of
two types namely
(, )
dy
f xy
dx
= (1)
or the Differential form
(, ) (, ) 0 P x y dx Q x y dy += (2)
where P and Q are functions of x and y.
These two equations clearly state that one of these forms may
readily be written in the other form.
For example the equation
1
()
dy x y
dx x y
++
=
-
is in the differential form
This equation can be written as
( 1) ( ) 0 x y dx y x dy ++ + - =
This is the other form of differential equation.
Similarly,
Institute of Lifelong Learning, University of Delhi pg. 3
Page 4
Solutions of First-Order Differential Equations
Discipline Course – I
Semester - II
Paper: Differential Equations - I
Lesson: Solutions of First-Order Differential
Equations
Lesson Developer: Anu Sharma
College: Aditi Mahavidalaya, University of
Delhi
Institute of Lifelong Learning, University of Delhi pg. 1
Solutions of First-Order Differential Equations
Table of Contents:
Chapter: Solutions of First-Order Differential Equations
• 1. Learning Outcomes
• 2. Introduction
• 3. Standard forms of First Order Differential Equations
o 3.1. Total Differential Function
o 3.2. Exact Differential Equation
• Exercise 1
• 4. Separable Equations
o 4.1. Solution of Separable Equations
• Exercises 2
• 5. Homogeneous Equations
o 5.1. Solution of Homogeneous Equations
• Exercises 3
• 6. Integrating Factors (I.F.)
• 7. Linear Equations
o 7.1. Solution of Linear Equation
o Exercise 4
o 7.2. Equations reducible to linear Form
o Exercises 5
• 8. Special Integrating Factors and Transformations
o 8.1. Rules for finding Integrating Factors
• Exercise 6
• 9. Special Transformations
o 9.1. Equations reducible to Homogeneous From
• Exercise 7
• Summary
• References
Institute of Lifelong Learning, University of Delhi pg. 2
Solutions of First-Order Differential Equations
1. Learning Outcome:
After reading this chapter, you should be able to :
(a) Recognize standard forms of First Order Differential Equations
(b)Exact Differential Equations and their solutions
(c)Separable Equations and equations reducible to this form and
their solutions
(d)Homogeneous Equations and equations reducible to this form
and their solutions
(e)Integrating Factors
(f)Linear Equations and Bernoulli’s Equations and their solutions
(g)Special Integrating Factors and Transformations
2. Introduction:
This chapter deals with first order differential equations for
which exact solutions can be obtained by definite methods. Here we
will discuss exact equations, separable equations, homogeneous
equations, linear equations and Bernoulli's equations. The aim of
this chapter is to recognize these various types of differential
equations and solve these by using various methods. Let us discuss
these equations one by one.
3. Standard forms of First Order Differential Equations
The differential equations of first order and first degree are of
two types namely
(, )
dy
f xy
dx
= (1)
or the Differential form
(, ) (, ) 0 P x y dx Q x y dy += (2)
where P and Q are functions of x and y.
These two equations clearly state that one of these forms may
readily be written in the other form.
For example the equation
1
()
dy x y
dx x y
++
=
-
is in the differential form
This equation can be written as
( 1) ( ) 0 x y dx y x dy ++ + - =
This is the other form of differential equation.
Similarly,
Institute of Lifelong Learning, University of Delhi pg. 3
Solutions of First-Order Differential Equations
the equation
(sin ) ( 3 ) 0 x y dx x y dy + ++ =
which is of the form
(, ) (, ) 0 P x y dx Q x y dy +=
can be written as
(sin )
( 3 )
dy x y
dx x y
+
= -
+
which is the differential form.
3.1. Total Differential Function:
Let U be a function of x and y, then the total differential dU of the
function U is defined as
(, ) (, ) U xy U xy
dU dx dy
xy
??
= +
??
Example 1: Let U be the function of two variables defined by
23
(, ) 2 U x y x y xy = +
Then find the total differential of the function U
Solution: The given function is
23
(, ) 2 U x y x y xy = +
Then
3
22
U
xy y
x
?
= +
?
(1)
and
22
6
U
x xy
y
?
= +
?
(2)
?
UU
dU dx dy
xy
??
= +
??
3 22
(2 2 ) ( 6 ) xy y dx x xy dy = + ++ [Using (1) and (2)]
3.2. Exact Differential Equation:
The differential equation
0 Pdx Qdy +=
is said to be exact if
Pdx Qdy dU +=
Institute of Lifelong Learning, University of Delhi pg. 4
Page 5
Solutions of First-Order Differential Equations
Discipline Course – I
Semester - II
Paper: Differential Equations - I
Lesson: Solutions of First-Order Differential
Equations
Lesson Developer: Anu Sharma
College: Aditi Mahavidalaya, University of
Delhi
Institute of Lifelong Learning, University of Delhi pg. 1
Solutions of First-Order Differential Equations
Table of Contents:
Chapter: Solutions of First-Order Differential Equations
• 1. Learning Outcomes
• 2. Introduction
• 3. Standard forms of First Order Differential Equations
o 3.1. Total Differential Function
o 3.2. Exact Differential Equation
• Exercise 1
• 4. Separable Equations
o 4.1. Solution of Separable Equations
• Exercises 2
• 5. Homogeneous Equations
o 5.1. Solution of Homogeneous Equations
• Exercises 3
• 6. Integrating Factors (I.F.)
• 7. Linear Equations
o 7.1. Solution of Linear Equation
o Exercise 4
o 7.2. Equations reducible to linear Form
o Exercises 5
• 8. Special Integrating Factors and Transformations
o 8.1. Rules for finding Integrating Factors
• Exercise 6
• 9. Special Transformations
o 9.1. Equations reducible to Homogeneous From
• Exercise 7
• Summary
• References
Institute of Lifelong Learning, University of Delhi pg. 2
Solutions of First-Order Differential Equations
1. Learning Outcome:
After reading this chapter, you should be able to :
(a) Recognize standard forms of First Order Differential Equations
(b)Exact Differential Equations and their solutions
(c)Separable Equations and equations reducible to this form and
their solutions
(d)Homogeneous Equations and equations reducible to this form
and their solutions
(e)Integrating Factors
(f)Linear Equations and Bernoulli’s Equations and their solutions
(g)Special Integrating Factors and Transformations
2. Introduction:
This chapter deals with first order differential equations for
which exact solutions can be obtained by definite methods. Here we
will discuss exact equations, separable equations, homogeneous
equations, linear equations and Bernoulli's equations. The aim of
this chapter is to recognize these various types of differential
equations and solve these by using various methods. Let us discuss
these equations one by one.
3. Standard forms of First Order Differential Equations
The differential equations of first order and first degree are of
two types namely
(, )
dy
f xy
dx
= (1)
or the Differential form
(, ) (, ) 0 P x y dx Q x y dy += (2)
where P and Q are functions of x and y.
These two equations clearly state that one of these forms may
readily be written in the other form.
For example the equation
1
()
dy x y
dx x y
++
=
-
is in the differential form
This equation can be written as
( 1) ( ) 0 x y dx y x dy ++ + - =
This is the other form of differential equation.
Similarly,
Institute of Lifelong Learning, University of Delhi pg. 3
Solutions of First-Order Differential Equations
the equation
(sin ) ( 3 ) 0 x y dx x y dy + ++ =
which is of the form
(, ) (, ) 0 P x y dx Q x y dy +=
can be written as
(sin )
( 3 )
dy x y
dx x y
+
= -
+
which is the differential form.
3.1. Total Differential Function:
Let U be a function of x and y, then the total differential dU of the
function U is defined as
(, ) (, ) U xy U xy
dU dx dy
xy
??
= +
??
Example 1: Let U be the function of two variables defined by
23
(, ) 2 U x y x y xy = +
Then find the total differential of the function U
Solution: The given function is
23
(, ) 2 U x y x y xy = +
Then
3
22
U
xy y
x
?
= +
?
(1)
and
22
6
U
x xy
y
?
= +
?
(2)
?
UU
dU dx dy
xy
??
= +
??
3 22
(2 2 ) ( 6 ) xy y dx x xy dy = + ++ [Using (1) and (2)]
3.2. Exact Differential Equation:
The differential equation
0 Pdx Qdy +=
is said to be exact if
Pdx Qdy dU +=
Institute of Lifelong Learning, University of Delhi pg. 4
Solutions of First-Order Differential Equations
or
UU
Pdx Qdy dx dy
xy
??
+= +
??
[by the def. of total differential]
or
and
UU
pQ
xy
??
= =
??
Theorem 1: Find the necessary and sufficient condition for the
equation Pdx Qdy += 0 to be exact.
Proof. (i) To find the necessary condition
Let Pdx Qdy += 0 be exact. (1)
Then by the definition of exact differential equation
, Pdx Qdy dU += where U is function of x and y. (2)
or
UU
Pdx Qdy dx dy
xy
??
+= +
??
(3)
Equating coefficients of dx and dy in (3), we get
U
P
x
?
=
?
(4)
and
U
Q
y
?
=
?
(5)
Differentiate (4) and (5) partially w.r.t. y and x respectively.
2
U
==
yx
P U
y yx
? ? ? ?
??
??
? ? ? ??
??
(6)
and
2
U
==
xy
QU
x xy
?? ? ?? ?
??
? ? ? ??
??
(7)
But
22
UU
y x xy
??
=
? ? ??
From (6) and (7), we get
=
PQ
yx
??
??
which is the required necessary condition.
(ii) To prove that the condition is sufficient
Institute of Lifelong Learning, University of Delhi pg. 5
Read More
FAQs on Lecture 2 - Solution of First Order Differential Equations - Differential Equation and Mathematical Modeling-II - Engineering Mathematics
| 1. What is a first order differential equation? |  |
Ans. A first order differential equation is a mathematical equation that relates an unknown function and its derivative. It only involves the first derivative of the unknown function.
| 2. How do you solve a first order differential equation? | |
Ans. To solve a first order differential equation, one can use various methods such as separation of variables, integrating factors, or exact differential equations. The specific method used depends on the nature of the equation.
| 3. What is the significance of solving first order differential equations in engineering mathematics? | |
Ans. Solving first order differential equations is crucial in engineering mathematics as it helps in modeling and understanding various physical phenomena. These equations are used to describe the behavior of systems in engineering, such as electrical circuits, fluid flow, and heat transfer.
| 4. Can you provide an example of a first order differential equation in engineering? | |
Ans. Yes, an example of a first order differential equation in engineering is the equation that describes the charging or discharging of a capacitor in an electrical circuit. This equation relates the voltage across the capacitor to the current flowing through it.
| 5. What are some applications of first order differential equations in engineering? | |
Ans. First order differential equations find applications in various fields of engineering. They are used to analyze and design control systems, model chemical reactions, simulate heat transfer in mechanical systems, and predict the behavior of electrical circuits, among others.
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